Buttered Cat Paradox
(Published on 5. January 2024, 19:50 by Kafkapharnaum)
All jokes aside, this is a tough one that assumes some high degree of insight into the implications of the ruleset, and so, if this ruleset is new to you, I recommend seeing beforehand if you can figure the two small warmup puzzles at the bottom of the page. Also if you enjoy this one or are just curious, there is other puzzles to explore in this series. – Huge thanks to wullemuus for testing this one dopplperschg doppltrgy
SOLVING LINKS (with tracking cells / no solution check)
Solve on the CTC app
Solve on Penpa
RULES
DOPPELGANGER
– Every row, column, and box contains the digit 0 always, as well as eight out of the nine digits from 1 to 9 (ie, it's missing one digit that is not 0)
– No two row, no two column, and no two box may be missing the same digit
– For each 0 in the grid, the digits missing in its row, its column, and its box must be three different digits (zeros always have a null value)
DOUBLERS
– Nine cells in the grid are doublers, which means that a digit in such a cell counts as double in value
– There is exactly one doubler in each row, column, and box, and they form a set of the digits 1 to 9 (so a doubler cell cannot contain a 0)
AMBIGUITY KILLER
– Each cage is either a 9 sum cage, or a single 9 cage, or both. A 9 sum cage contains digits whose values sum to 9. A single 9 cage contains a single cell with the value 9 – ie, an undoubled digit 9
– Values can repeat within a cage but digits may not. Eg, where * means a doubler, 9-4*-8-0-3 would be a valid cage, but 4*-9-7-1-4 wouldn't, regardless of the 9
– A cage cannot contain multiple doublers
Warmup puzzles (follow the links to find the rules for those)
6x6
4x4
Solution code: The doublers, in order when reading the grid from top to bottom (ie, row after row)
Comments
on 24. February 2024, 03:12 by gfoot
I wanted to say, having done some other doppelganger puzzles recently, I decided to come back to this one that has really stuck in my mind, and solve it again, which was very rewarding as I was able to experiment with different ways through, and find simpler and more direct deductions than I did the first time. It was thoroughly enjoyable again, you have done a really great job here.
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Wow, thank you so much for that very heartwarming comment, I'm touched that you liked that puzzle enough to revisit it and then take the time to write this!
This is all the more pleasing that this is the puzzle that took me the longest to conceptualize, as I'd had this idea of the Buttered Cat Paradox with 9 sums vs plain 9's for well over a year and a half, and failed multiple times to come up with anything out of it, and I was starting to think I wouldn't be able to right up to the point where I finally pulled through with this one
And yet, believe it or not, this still is not quite the puzzle that I'm the proudest of, as this would still be Solaris for the sheer amount of raw efforts that this one took me, and seeing you so well disposed toward the ruleset, I can't help but feel compelled to recommend it! But in any case, thank you again so much for the support! :D
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Yes I really like the way the ambiguity in the cages blends with possible doublers, zeros, and unknown missing digits, all of these ambiguities together make it challenging to make progress but you get there bit by bit. Are Ambiguity Cages a novel concept? I was solving Ancient Tablets, and will try Solaris now.
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You're crushing these! Congrats, and thanks so much for the support :D
Regarding your question, well, there's been other types of ambiguities tried with cages, but not that precise one of 9 sums vs plain 9's -- there was just something that seemed impossible about it, without bringing in some other type of clues, and indeed, as you can see, I had to bring in doublers, as I'm pretty sure that even the powerful Doppelganger ruleset still wasn't enough o its own (would love to see that proven wrong though!), and that was my missing piece in finally creating this one
Good luck on Solaris, and thanks so much again! :D
on 9. February 2024, 02:42 by bodemeister
Hard but rewarding. A lot to consider at nearly every step! Thanks for the puzzle!
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Whoa, what an awesome surprise, thank you so very much! Clealy this ruleset doesn't have any secrets for you XD Glad you enjoyed!
on 13. January 2024, 18:03 by JayForty
Really enjoyed this puzzle! Extremely clever (and difficult). It definitely helped doing the warm-up puzzles. That 4x4 is a real gem, absolutely amazing logic in such a small puzzle! Thanks!
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Thank you so much for your solve and for such kind words! Any solve on this one is bound to make my day, and I'm all the happier to see that after being a bit worried that the warmup puzzles could have been maybe better selected, that they seem to be fulfilling their purpose just fine after all :D
I'm thrilled you enjoyed the whole lot like that, thanks so much again!
on 9. January 2024, 16:58 by wullemuus
It was a great pleasure to test this puzzle with one of my favourite ruleset! The combination of ambiguity killers, doublers and zero allows a lot of unexpected variants. Absolutely worthwhile like all the others!
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Thank you so much for entering your solve, and for your comment! I can't think of a better way that my 2023 puzzle year could have ended than by you sending me the solution to this one on Dec 31st all the while wishing me a Happy New Year haha, that was amazing! Thanks once more for your tremendous support :D
on 7. January 2024, 02:52 by gfoot
Oh wow, what a puzzle. I am not the best solver and it was very challenging, it took several hours to solve and was very intricate and well put together.
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Thank you so much for your perseverance in breaking the ice on this one, and for the kind words, I'm thrilled to hear you enjoyed it!
As with all the others of these puzzles (those from the linked list), any solve on these is really special to me, so you really made my day there! :D
And it's always an extra pleasing surprise when it's from people whom, as far as I know, had never solved one of these before -- did you solve any of the warmup puzzles beforehand, or did you just jump in fully blind?
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No I haven't solved any of the others in tge series, nor the warm-ups - I'd already read the cage rules, and thought a warm-up based on other logic might just confuse me!
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Ahh, that's a good point, and maybe I shouldn't have been lazy in just selecting older puzzles, and rather created a more bespoke example. Anyhow, that's pretty impressive, as it must have (I imagine) added a number of extra deductions that I just 'took for granted', so to speak, so thanks all the more for your solve!
